Selectivity Options for Age-Structured Stock Assessments
A comparative framework using RTMB
Selectivity is a fundamental component of age-structured stock assessment models, governing how fishing mortality and survey catchability vary with age or size. The choice of selectivity parameterization affects estimates of population abundance, fishing mortality, and management reference points. This manuscript compares several selectivity formulations—including standard logistic, double logistic, spline-based, and time-varying 2D autoregressive approaches—within a unified RTMB framework. We examine parameter identifiability, prior sensitivity, and the practical consequences of selectivity misspecification using simulation and application to eastern Bering Sea walleye pollock (Gadus chalcogrammus). Results highlight trade-offs between flexibility and estimability and provide guidance for practitioners choosing among selectivity options in operational assessments.
selectivity, stock assessment, RTMB, fisheries, walleye pollock
0.1 Introduction
Selectivity functions describe the relative vulnerability of fish to capture as a function of age (or size) and are among the most influential—yet least observable—components of stock assessment models. The assumed shape of the selectivity curve directly affects estimates of spawning biomass, recruitment, and fishing mortality, and misspecification can propagate into biased reference points and management advice (Punt, Hurtado-Ferro, and Whitten 2013; Thompson 1994).
Despite its importance, selectivity is often treated as a modelling convenience rather than a biological or technological quantity to be estimated carefully. Most operational assessments adopt a logistic or double-logistic functional form, chosen for parsimony rather than fidelity to the underlying catch process. More flexible alternatives—such as penalized splines or non-parametric approaches—can reduce structural bias but may introduce identifiability problems, particularly when data are sparse or conflicting.
This manuscript develops a comparative framework for evaluating selectivity options within the R Template Model Builder (RTMB) environment. RTMB provides automatic differentiation, Laplace approximation for random effects, and integration with Bayesian sampling via SparseNUTS, making it a natural platform for exploring both maximum likelihood and posterior-based inference on selectivity parameters.
We organize the comparison around four selectivity families:
- Standard logistic (Section 0.2): the two-parameter ascending logistic, widely used for fisheries where retention is assumed to be monotonically increasing with age.
- Double logistic (Section 1.5): a dome-shaped three-parameter formulation allowing selectivity to decline at older ages, motivated by gear avoidance, ontogenetic habitat shifts, or differential availability.
- Spline-based (Section 2.9): a penalized B-spline approach offering flexible, data-driven selectivity shapes with smoothness controlled by a penalty parameter.
- Time-varying 2D AR1 (Section 3.6): a separable autoregressive structure over year and age dimensions, allowing selectivity to evolve smoothly through time while maintaining age-specific coherence via a Kronecker-structured precision matrix.
For each family, we present the mathematical formulation, an RTMB implementation with prior specification, parameter sensitivity analysis, and MCMC-based posterior evaluation. We then apply all to eastern Bering Sea walleye pollock data to illustrate practical trade-offs.
0.2 Standard Logistic Selectivity
1.5 Double Logistic Selectivity
2.9 Spline-Based Selectivity
3.6 Time-Varying Selectivity with 2D Autoregressive Structure
4.7 Comparison and Discussion
Placeholder: comparative analysis across selectivity families, including AIC/BIC, posterior predictive checks, and implications for management quantities.